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Reynolds ( ) Ernst Mach (1838 – 1916)
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SIGNIFICANT DIMENSIONLESS GROUPS
IN FLUID MECHANICS Inertia Viscous Pressure FORCES: (in Fluid Mechanics) Gravity Surface Tension Compressibility
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PHYSICAL MEANINGS OF DIMENSIONLESS GROUPS DIMENSIONLESS GRUOPS Inertia
Reynolds Number (Re) Viscous Euler Number (Eu = Cp) Pressure DIMENSIONLESS GRUOPS Froude Number (Fr) Weber Number (We) Mach Number (M) Gravity Surface Tension Compressibility
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FLOW SIMILARITY AND MODEL STUDY
EXPERIMENTAL DATA To calculate secondary data: Must be Scaled - Forces - Moments, etc. There must be similarity between MODEL and PROTOTYPE - Similar in shape - Constant (linear) scales - GEOMETRIC SIMILARITY: - KINEMATIC SIMILARITY: - DYNAMIC SIMILARITY: - Similar flow kinematics - Constant scales (in magnitudes) - All forces scaled constantly - Needs geometric & kinematic sim’s.
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FLOW SIMILARITY AND MODEL STUDY (Cont’d)
For complete analysis All contributing forces must be presented: - Viscous force - Pressure force - Surface tension force - etc. Buckingham p – theorem can be used
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Drag force analysis on a sphere: F = f (D, V, r, m)
EXAMPLE Drag force analysis on a sphere: F = f (D, V, r, m) F r, m, V D From p – theorem, we have: The flow will dynamically similar if: Also: Similarity in the ratio of drag to inertia forces between model & prototype Similarity in the ratio of inertia to viscous forces between model & prototype
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Sonar transducer model tested in a wind tunnel
EXAMPLE PROBLEM 7.4 Given: Sonar transducer model tested in a wind tunnel Find: a) Vm b) Fp Solution: The test should be run at:
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EXAMPLE PROBLEM 7.4 5.02 x 105 Therefore: 5.02 x 105 And:
Vm = 156 ft/sec m/s Finally: Fp = 54.6 lbf
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